Exotic fermionic fields and minimal length
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Regular Article - Theoretical Physics
Exotic fermionic fields and minimal length J. M. Hoff da Silva1,a , D. Beghetto1,2,b , R. T. Cavalcanti1,c , R. da Rocha3,d 1
Departamento de Física, Universidade Estadual Paulista, UNESP, Av. Dr. Ariberto Pereira da Cunha, 333, Guaratinguetá, SP, Brazil Instituto Federal de Educação, Ciência e Tecnologia do Norte de Minas Gerais, IFNMG, Av. Dr. Humberto Mallard, 1355 Pirapora, MG, Brazil 3 CMCC, Federal University of ABC, 09210-580 Santo André, Brazil
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Received: 5 June 2020 / Accepted: 30 July 2020 © The Author(s) 2020
Abstract We investigate the effective Dirac equation, corrected by merging two scenarios that are expected to emerge towards the quantum gravity scale. Namely, the existence of a minimal length, implemented by the generalized uncertainty principle, and exotic spinors, associated with any non-trivial topology equipping the spacetime manifold. We show that the free fermionic dynamical equations, within the context of a minimal length, just allow for trivial solutions, a feature that is not shared by dynamical equations for exotic spinors. In fact, in this coalescing setup, the exoticity is shown to prevent the Dirac operator to be injective, allowing the existence of non-trivial solutions.
1 Introduction Over the last decades, high energy physics has allowed us to look closer to the very structure of matter. It naturally rises concerns on the limits of current approaches. Whether this limit exists or not, fundamental concepts regarding the underlying spacetime geometry may be revisited, at high energy scales [1,2]. At least at energy scales as large as the Planck scale, quantum gravity effects are expected to set in. The non-renormalizability of quantum gravity consists of a major problem that physicists have been trying to overcome for decades. However, it could be effectively circumvented, by suggesting that gravity should lead to an ultraviolet cutoff, leading to a minimal observable length. The existence of a minimal length scale seems to be a model-independent feature of quantum gravity [3], from string theory [4–6] to loop quantum gravity [7], and quantum black holes [8–13]. Effectively, the Heisenberg Uncertainty Principle (HUP) can be corrected by a minimal length, giving rise to the so-called a e-mail:
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Generalized Uncertainty Principle (GUP). See Refs. [14–17] for a comprehensive review. In particular, in the context of the Dirac equation, phenomenological bounds were explored in Ref. [18]. Formally, the minimal length can be implemented in quantum mechanics, through small quadratic corrections to the canonical commutation relations [19–22], whose relationship to a minimal length is not biunivocal [23]. The compatibility with relativistic covariance was implemented in Ref. [24], with the so-called Quesne-Tkachuk algebra, leading to a generalization of the Heisenberg alge
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